Non-formal deformation quantization of abelian supergroups

TL;DR

This paper reviews recent advances in deformation quantization of abelian supergroups, focusing on representation construction, star-products, and applications in supergeometry and physics.

Contribution

It introduces a new deformation quantization framework for abelian supergroups using pseudodifferential calculus and Hopf algebra techniques.

Findings

Constructed an induced representation of the Heisenberg supergroup.

Developed a star-product and universal deformation formula for abelian supergroups.

Introduced the concept of C*-superalgebra compatible with deformation.

Abstract

We review recent works concerning deformation quantization of abelian supergroups. Indeed, we expose the construction of an induced representation of the Heisenberg supergroup and an associated pseudodifferential calculus by using Kirillov's orbits method. Then, a star-product is built on the abelian supergroup R^{m|n} together with a universal deformation formula for its actions. Using topological Hopf algebras, we reformulate this deformation as a continuous twist on comodule-algebras. We also introduce the notion of C*-superalgebra which is natural and compatible with the deformation. Finally, we show some applications in supergeometry and theoretical physics.